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Crystal Structures Crystal Structures D. K. Ferry, J. P. Bird, D. Vasileska and G. Klimeck Crystal Structures — Matter comes in many different forms the three we are most used to are solid, liquid, and gas. water ice Air Water vapor etc. Matter in its Many Forms Matter in its Many Forms – Heating ice produces water (T > 0 C, 32 F melting point) Heating water produces steam (water vapor) (T > 100 C, 212 F) These transitions are phase transitions. Some materials have triple points, where all three forms can exist together at the same time. 1 1 density density gas gas liquid liquid triple point solid solid T T Matter in its Many Forms • In LIQUIDS the density of molecules is very much higher than that of gases * BUT the molecular interaction is STILL relatively weak NO structural stability • SOLIDS may have a similar molecular density to that of liquids * BUT the interaction between these molecules is now very STRONG The atoms are BOUND in a correlated CRYSTAL structure Which exhibits MECHANICAL STABILITY IN SOLID MATTER THE STRONG INTERACTION BETWEEN DIFFERENT ATOMS MAY BIND THEM INTO AN ORDERED CRYSTAL STRUCTURE Classifying Solids • An issue concerns the classification of ORDER in crystal materials * In order to achieve this we can define the RADIAL DISTRIBUTION FUNCTION Which tells us WHERE we should find atoms in the material This parameter can be measured experimentally P(r) GAS OR LIQUID r HIGHLY DISORDERED CRYSTAL HIGHLY ORDERED CRYSTAL P(r) IS THE POSITION AVERAGED PROBABILITY OF FINDING ANOTHER ATOM AT A DISTANCE r FROM SOME REFERENCE ATOM r Classifying Solids A transmission electron microscope (TEM) image of a highly ordered crystal of Si. We will learn soon about how to discribe the different directions in such a crystal as this. Classifying Solids--Crystals • In reality very FEW materials exhibit perfect crystalline order * Most materials develop IMPERFECTIONS during their fabrication process Unwanted IMPURITY atoms POINT and LINE DEFECTS Thermodynamics and ENTROPY LINE DEFECTS IN SiGe VACANCY INTERSTITIAL DEFECT EXAMPLES OF LINE DEFECTS Classifying Crystals • A large number of materials exhibit POLYCRYSTALLINE order * These materials consist of regions with NEARLY PERFECT order That are separated by thinner DISORDERED regions The RELATIVE orientation of the CRYSTALLITES is RANDOM DISORDERED Si CRYSTALLINE Si SCHEMATIC ILLUSTRATION OF TRANSMISSION ELECTRON MICROGRAPH POLYCRYSTALLINE ORDER SHOWING A 10 nm Si CRYSTALLITE EMBEDDED IN DISORDERED Si Classifying Crystals • Other materials are referred to as AMORPHOUS * These are solids in which the molecules are almost RANDOMLY arranged LITTLE or no long range order exists These materials are essentially HIGHLY VISCOUS liquids GLASS is an example of an amorphous solid A PERFECT CRYSTAL AN AMORPHOUS ATOMIC STRUCTURE ARRANGEMENT Classifying Solids Transistors in our Integrated Circuit ’ Let s look closer in here Classifying Solids Transistors in our Integrated Circuit SiO2 is an amorphous material which is used for the insulator--it grows naturally on Si at high temperature. Crystalline Si is used as the base material for all processing in the integrated circuit world. Poly-crystalline Si, or a metal such as Al, is “ ” used to make the gate of the transistor (we will learn later just what the gate does). Preparing the Silicon for VLSI • Most materials may actually exist in ALL three different states * Depending on the manner in which they were FABRICATED * Crystalline materials are the most DIFFICULT to make And typically require special processes SINGLE CRYSTAL SILICON IS GROWN BY THE CZOCHRALSKI METHOD AN 8-in DIAMETER INGOT OF SINGLE CRYSTAL SILICON IS SHOWN ABOVE Now, crystals can be made in many shapes. You can even do it at home. — Silicon crystals have to have very high purity no defects, vacancies, or random non- silicon atoms. This is done via the CZOCHRALSKI METHOD. A special machine for the vertical growth of crystals via the Czochralski method. Preparing the Silicon for VLSI A very thin wafer of the ingot is used to process and create the integrated circuit. More than 200 individual chips will be made on each wafer. ” Five years ago: We used 8 wafers. This Year: The Intel facilities in ” Chandler all use 12 wafers! Classifying Materials • Not all materials are easily classified using the schemes described above * Liquid crystals are HIGHLY VISCOUS liquids containing LONG molecules With STRONG and PERMANENT electric dipole moments * In the absence of an electric field the molecules are RANDOMLY oriented But an electric field may be used to ALIGN their dipoles IN THEIR UNALIGNED STATE WE MAY PICTURE THE MOLECULES OF THE LIQUID CRYSTAL ARRANGED AS THE LOGS IN THIS PHOTOGRAPH – Covalent bonds the key to life (and semiconductors)! Highly directional 4 hybrids of s- and p- electrons form Since the hybrids are composed of – electrons, they repel each other most uniform result is tetrahedral shape Array of Atoms in Diamond Lattice … (diamond, silicon, germanium, grey tin, ) Group IVB: 4 outer electrons 1 s-electron, 3-p electrons Compounds from these two groups can form with an average of 4 electrons per atom. GaAs GaP InP AlAs InAs GaN Compounds from these two groups can form with an average of 4 electrons per atom. Crystalline Order The high degree of order in this TEM picture of Si is apparent. [http://cst-www.nrl.navy.mil/lattice/struk/a4.html] The order is more apparent when we understand how this picture is made through a process known as lattice plane imaging. Diffraction of the electrons occurs from each plane of atoms; the diffracted beams of electrons form a Fourier transform in the microscope, and this is re- transformed to give the image. For those interested, a review of lattice plane imaging technology is found at: http://www.xray.cz/ms/bul2001-1/karlik.pdf First, however, not everything is crystalline: Crystalline Order • We have seen that crystalline systems may exhibit varying degrees of ORDER * In AMORPHOUS materials there is little order at all * In POLYCRYSTALLINE materials the order is INTERMEDIATE Poly-crystalline grain of Si embedded in amorphous Si • • In many cases though it is convenient to assume the crystalline order is PERFECT THIS ATOMICALLY PRECISE SANDWICH OF GaAs SEMICONDUCTOR MATERIALS (GaAs & AlAs) EXHIBITS HIGH CRYSTALLINE ORDER AND WAS GROWN BY A TECHNIQUE KNOWN AS MOLECULAR BEAM EPITAXY. NOTE THE ALMOST PERFECT ARRANGEMENT OF INDIVIDUAL ATOMS IN THIS FIGURE AlAs This precision is achievable because both GaAs and AlAs have the same lattice constant. Å 3 By crystal structure, we mean the regular (or not) arrangement of the atoms. “ ” We can talk about planes of atoms (which are rows in this 2D projection of the 3D lattice. [http://cst- www.nrl.navy.mil/lattice/struk/a4.html] These planes define directions — relative to the planes each plane has a surface normal vector which defines the plane. aN Describing Crystal Structures How do we identify various crystals, and tell two materials apart from their crystal properties? We need a method by which we can describe the crystal in terms of its basic properties. This can be done in one sense, because there are only a few types of crystal lattice. Hence, identifying this structure tells us a lot about the crystal. Describing Crystal Structures • An IDEAL crystal is an INFINITE repetition of IDENTICAL structural units in space * These units may range from single atoms to complex molecules • We therefore need TWO important quantities to DESCRIBE a given crystal structure * A multi-dimensional LATTICE: a set of POINTS in space * A suitable BASIS: the unit that will be placed at EACH lattice point + = 2-D LATTICE BASIS 2-D CRYSTAL “ ” The basis may be a single atom, or a small group of atoms. Describing Crystal Structures • The crystal LATTICE may be defined using appropriate TRANSLATION VECTORS * If we know where ONE lattice point is situated in the crystal These vectors determine the position of OTHER lattice points • STARTING FROM AN ARBITRARY ORIGIN O WE MAY USE TRANSLATION VECTORS a & b TO MAP TO ANY r r` r`` OTHER LATTICE POINT: r 2b b r' a 2b r'' 3a 2b O a * Here a and b are referred to as PRIMITIVE translation vectors * Since they can be used to determine the position of ALL lattice points Describing Crystal Structures * A 2-D crystal structure consists of a rectangular lattice with primitive translation vectors a and 2a in the horizontal and vertical directions. The basis unit of this crystal consists of two different atoms that are placed at (a/2)[x, y] and (a/2)[x,-y] relative to each lattice point. Draw the crystal. a + = 2a BASIS LATTICE “ ” CRYSTAL STRUCTURE We have to use a basis because the ’ atoms themselves don t form a proper lattice Typical 3D Lattice Types • While the number of materials exhibiting crystal order is very LARGE in nature * The number of lattice TYPES encountered is actually quite small * Here we discuss the most important lattice types found in nature • Real crystal structures are THREE-DIMENSIONAL in nature * In a SIMPLE CUBIC lattice the three primitive vectors have EQUAL length • REMEMBER THIS IS JUST THE LATTICE! • THE BASIS OF ATOMS WE ADD TO EACH LATTICE POINT DETERMINES WHAT MATERIAL WE FINALLY END UP WITH • FOR EXAMPLE, ADDING MANGANESE ATOMS TO EACH LATTICE POINT WOULD GIVES CRYSTALLINE MANGANESE Typical 3D Lattice Types • The BODY CENTERED CUBIC lattice is formed by adding an ADDITIONAL lattice point * To the CENTER of each cube defined by the simple cubic lattice * This lattice may be considered as TWO interlocking simple cubic lattices Typical 3D Lattice Types For the BODY CENTERED CUBIC lattice, we must be careful to define the primitive translation vectors to include the new atom at the center of the cube. Each direction is rotated to a new body-centered atom.
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